Game Theory in Aperiomics

Aperiomics is a system I thought of in 1989, I’ve been working on it mainly privately since then but am now starting to publish it. More detailed of it are found at Aperiomics.org, it is based on 12 mathematical principles of chaos and randomness that combine to explain events in war, economics, crime, sociology, evolution, etc.
People are welcome to read, they can correspond with me at greg@aperiomics.org.

Monday, December 17, 2012

Evolutionary game theory (EGT) is the application of game theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies, and analytics into which Darwinian competition can be modelled.

Aperiomics is something similar, however it models several different kinds of competition and cooperation.

The need for evolutionary game theory in biology started with a problem. The problem was how to explain ritualized animal behaviour in a conflict situation; "why are animals so "gentlemanly or ladylike" in contests for resources?" This was a problem that leading ethologists Niko Tinbergen and Konrad Lorenz were trying to address.

Often the environment is a Biv situation where resources are plentiful, it is then more efficient for animals to evolve politeness and look for resources elsewhere. In a Roy environment resources are scarce and so this politeness is less common, for example predators in an African drought.

Tinbergen proposed that such behaviour exists for the benefit of the species. Maynard Smith couldn't see this reasoning matched with Darwinian thought as he understood it,[3] where selection occurs at an individual level and therefore defections to self-interest are rewarded while seeking the common good is not.

In some cases, however in Roy there are Y and Oy predators attacking Ro and R prey. When a team of Y predators such as lions attacks prey they often find it is safer to also defend as a team. In the Roman Empire the legions would usually attack as a Y team, the idea was to make the defensive Ro army break ranks and scatter so they would be easy to pick off one by one. In WW2 the Y German tactic was often to split up parts of the Ro defensive armies and then encircle them, the whole Y army could then overwhelm these small pockets.

The point then is not that self-interest is better or worse than seeking the common good but which strategy the opposition is using.

Oy predators work as individuals, when they attack a Ro herd like hyenas attacking buffalo they need to pick off older or weaker animals. In this situation then teamwork defeats the individual Oy attackers. When Y lions attack R gazelles the gazelles scatter and hide, often this self-interest defeats the common good behaviour of Y lions.

On reflection Maynard Smith realised that in an evolutionary version of Game Theory it is not really a requirement that players be rational – it is only required that they have a strategy. The results of the game will test how good that strategy is. That is what Evolution does – it tests alternative strategies for the ability to survive and reproduce. In Biology, strategies are genetically inherited traits that control an individuals action – and strategies are algorithmic – just like computer programs. The key point in the Evolutionary Game Theory model is that the success of a strategy is not just determined by how good the strategy is in itself, it is a question on how good the strategy is in the presence of other alternative strategies, and on the frequency that other strategies are employed within a competing population. It is also a question of how good a strategy plays against itself, because in the biological world a successful strategy will eventually dominate a population and competing individuals in it end up facing identical strategies to their own.[5]

In Aperiomics it is also not necessary to have a strategy at all, Oy-R and Iv-B interactions are tactics versus tactics where participants chaotically do whatever works best on the margin. For example an Oy fox doesn't need a strategy to hunt for R rodents, it just needs to catch them when it sees them. Some animals have a strategy in breeding, for example Ro herds of buffalo might have a planned or ritualistic system for which males get to breed. Other animals such as R rodents might just mate with whatever rodent of their species they come across.

Each system develops different kinds of animals, the Ro herd by planning its system of breeding reinforces cooperative genes. Those that try to push in or break up this ritual are forced out by the team, otherwise the ritual might collapse and the Ro herd becomes R competitors. In this case it is not in most of the male's interest to cooperate and instead they each try to mate when they can, even deceptively and secretly. For example Oy hyena females or R meerkats might mate with an outside male secretly. The genes then become more chaotic in what they produce in offspring, this can lead to R revolutionary or Oy counter revolutionary changes in these animals. For example in a Ro herd the planned system of males that win a ritual contest leads to them mating and more males willing to submit to this contest.

With chaotic R breeding any instinct for ritualistic mating gets broken up by competing instincts from other genes. This is also seen in humans, some societies have a highly structured contest where male suitors try to win females, others have males deceptively cuckolding married women. A Ro or Bi planned mating system leads to cooperative behaviour and better defences against Y or V adversaries. In a V part of society the talented elite also have these cooperative rituals such as prom dances, dating, marriage, etc. They also have some Oy competition breaking up these rituals such as marrying out of wedlock, polygamy, and affairs.

The object of the evolutionary game is to become more fit than competitors – to produce as many replicas of oneself as one can and the payoff is in units of Fitness (relative worth in being able to reproduce).

In Aperiomics there are two games, an evolutionary game where the object is to become more normal in a cooperative system, and a revolutionary game where the tactics are to win often by deception and secrecy. The payoff in this cooperative Y-Ro or V-Bi game need not be fitness except in a tautological sense, it can also be to avoid being on the fringes of a team where it is more dangerous. For example more normal Ro buffalo might support each other more while more deviant prey end up on the edges eaten by predators. This is seen for example where some team animals shun offspring that are mutated, these might be revolutionary gene combinations that are weeded out to maintain normality and evolution in incremental changes. Oy-R and Iv-B however use these revolutions to grow and sometimes collapse chaotically.

The payoff for these competitions is to win on the margin, this need not mean they are more fit but they can also be better at swindling their opponent. For example R gazelles might be weak and unfit but can survive by hiding and using camouflage. Often less fit animals can breed by stumbling on a mate without a strategy, they are available and then offspring come from that rather than from fitness. This fitness implies normality as a standard, however chaotic competition has no normality and revolutionary abilities can lead to more offspring.

It is always a multi-player game with a very large population of competitors.

In the chaotic Oy-R and Iv-B colors the game is usually between small numbers of individuals where the other animals are largely irrelevant and not part of their tactics.

Rules describe the contest as in classical Game Theory but for evolutionary games rules include the element of replicator dynamics, in other words the general rules say exactly how the fitter players will spawn more replicas of themselves into the population and the less fit will be culled out of the player population (expressed in a Replicator Equation).[6] The replicator dynamics in essence models the heredity mechanism, but for simplicity leaves out mutation.

However in chaotic Oy-R and Iv-B mutation is a vital part of their tactics, this often creates new species or animals with new abilities to either catch prey or evade predators.

Similarly, Evolutionary Game Theory only uses asexual reproduction for the sake of simplicity. Games are run repetitively with no terminating conditions.

There are often terminating conditions in Oy-R and Iv-B, mutated animals might be wiped out, also animal populations tend to boom and bust such as with R locusts exploding in numbers and then collapsing for a lack of food. Also sexual reproduction can be highly chaotic as males might deceptively overwhelm a female or in the case of Oy hyenas the females can choose males outside their team to increase the variations of their genes. So female hyenas can see a male that is abnormal compared to their team but has counter revolutionary abilities that are appealing, for example different camouflage markings or able to run faster.

Sexual reproduction in Y-Ro and V-Bi tends to follow rituals so the winner spreads his genes to most of the females, this makes them more related to each other and less likely to compete chaotically with or leave behind to predators a blood relative.

The results that are studied include the dynamics of changes in the population, the success/survival of strategies and any equilibrium states reached in competing.

In Aperiomics there are generally no equilibrium states in competition, instead there are chaotic changes like fractals. Equilibrium happens around normality in Y-Ro and V-Bi.

Unlike classical Game Theory players do not choose their strategy or have the ability to change it, they are born with that strategy and their offspring will inherit that same identical strategy.

In chaotic Oy-R and Iv-B there is no strategy, offspring however can use different tactics. For example R gazelles might have offspring with mutated camouflage markings, this dictates their tactics of running or hiding when there are predators nearby.

Some representative games of Evolutionary Game Theory are hawk-dove, war of attrition, stag hunt, producer-scrounger, tragedy of the commons, and prisoner's dilemma. Some of the various strategies that apply in these games are Hawk, Dove, Bourgeois, Prober, Defector, Assessor, and Retaliator. Depending on the particular “Game” the various strategies vie against one another under the particular game rules, and the mathematics of the evolutionary game theory are used to determine the results and behaviours.

The game structure of Aperiomics is explained in the Game Theory chapter of my book Micraperiomics at aperomics.org. While this refers to people the interactions are the same as for animals and plants, how the colors refer to these is also explained in the book.

Hawk Dove

The most classic game (and Maynard Smith's starting point) is the Hawk Dove game. The game was conceived to analyse the animal contest problem highlighted by Lorenz and Tinbergen. It is a contest over a sharable resource, say some food. The contestants can be either Hawk or Dove… this is not two separate species of bird; it is ONE species with two different types of strategy in the same species (two different morphs). The term Hawk Dove was coined by Maynard Smith because he did his work during the Vietnam War when political views fell into one of these two camps. The strategy of the Hawk (a fighter strategy) is to first display aggression, then escalate into a fight till he either wins or else is injured. The strategy of the Dove (fight avoider) is to first display aggression but if when faced with major escalation by an opponent will run for safety. If not faced with this level of escalation the Dove will attempt to share the resource.

In this game the hawk can be like Y or Oy, it might share food with a Y cooperative strategy or try to compete as Oy. The dove might be in a flock with others, when faced with aggression it might stay together and defend such as with a small hawk. If the hawk is too big they might split up with an R chaotic strategy to run and hide.

The actual payoff however depends on the probability of meeting a Hawk or Dove, which in turn is a representation of the percentage of Hawks and Doves in the population when a particular contest takes place.

In Aperiomics this is not just probability, it can also be a chaotic tipping point where an Oy hawk might fight until it becomes too dangerous and then suddenly runs from the stronger hawk. When chaotic the odds of meeting an Oy hawk or R dove might be unrelated to the percentage in the general population, this is because they move so as not to be anticipated. R doves are then likely to be found in unexpected places and numbers when there is danger, also Oy hawks would be concentrated in areas unrelated to a normal curve.

But that population makeup in turn is determined by the results of all of the previous contests before the present contest- it is a continuous iterative process where the resultant population of the previous contest becomes the input population to the next contest.

This can also be chaotic or fractal like where the output of one deterministic equation feeds back into itself or into a related equation. This is not the same as random probability such as where the spins of a roulette wheel cumulate to particular fractions of numbers appearing. For example instead of a hawk probably winning according to their population ratio the contests might vary chaotically from boom to bust, as hawks win more then the doves get less food and so decline in numbers or move elsewhere for food. If the hawks feed on the doves then as the dove population crashes the hawks will begin to starve thinning out their numbers allowing for a rebound of the dove numbers.

If the cost of losing C is greater than the value of winning V (the normal situation in the natural world) the mathematics ends in an ESS – an evolutionary stable strategy situation having a mix of the two strategies where the population of Hawks is V/C. The population will progress back to this equilibrium point if any new Hawks or Doves make a temporary perturbation in the population. The solution of the Hawk Dove Game explains why most animal contests involve only “ritual fighting behaviours” in contests rather than outright battles. The result does not at all depend on “good of the species” behaviours as suggested by Lorenz, but solely on the implication of actions of “selfish genes”.

This stable strategy only happens when the Y hawks and Ro doves each act like rival teams, then probability can predict when each might eat. This is a cooperative strategy and leads to ritual fighting behaviours rather than because of selfish genes. Instead the genes themselves act altruistically because they survive better as do some animals.

For example the genes in a Ro herd of buffalo survive because they tend to be more normal, those more deviant genes get culled out of the herd by Y and Oy predators taking animals on the edge of the herd. A buffalo that takes too long to grow up and walk with the herd then is abnormal and its genes are culled by predators. Competitive or selfish genes move between booms and busts in innovations, because they are selfish they are in effect trying to replicate at the expense of other genes in the Ro herd.

This selfishness then makes the Ro herd more diverse and opens up more chaotic cracks in their defences that predators can exploit, this is like a Ro army where differences in the strength or fighting ability of soldiers can indicate a weakness to exploit. Once this weakness is defeated it is then like a wedge into the Ro position allowing it to be split or shattered.

There are then in Aperiomics competitive and cooperative genes, or selfish and altruistic genes. For example a Ro herd of Ibex might have some that are smaller and faster while others are slower and stronger. If their genes are competitive then they might scatter and hide as R from predators, the more chance of another Ibex getting eaten the better the outcome for a selfish gene. So a faster Ibex might evolve if the Oy predators are faster than the slow Ibex, however a stronger Ibex might evolve is the Oy predators are only strong enough to bring down the fast Ibex. For example there might be wild dogs in Africa not strong enough to bring down the slower Ibex.

Instead of competing as R these genes might cooperate as Ro to minimize their losses form predators so there end up being more of these genes in circulation in more Ibex. Instead of splitting up when attacked by Oy single predators the Ibex stand together as a Ro team driving Oy away, two Ibex might always win as a team like a prisoner's dilemma but one might die is they scatter. As some Ibex try this Ro cooperative strategy then they mate with each other more, this is instead of genes being selfish and only mating with those like it so the inferior prey are eaten instead. The result is a more normal Ro Ibex that is an average speed and strength whose genes survive better because they defend each other. These genes however might have to become R chaotic again later if Y predators are too strong for the Ibex, the cooperative gene strategy would then fail and the Ibex would split up and hide. This is then why there are R prey such as gazelles and Ro prey such as buffalo or Wildebeest.

The War of Attrition game

In the Hawk Dove game the resource is sharable, which gives payoffs to both Doves meeting in a pairwise contest.

This can allow them to work as a Ro team, for example if there is enough food for 5 doves to share then they might collectively chase away a lone Oy Hawk.

In the case where the resource is not sharable but an alternative resource might be available by backing off and trying elsewhere, pure Hawk or Dove strategies become less effective.

Ro teamwork here is less effective because they might get more food by splitting up rather than competing over one area. It can then be a Biv area where food is more abundant so the animals use a Profit maximizing strategy instead of loss minimization.

If an unshareable resource is combined with a high cost of losing a contest (injury or possible death) both Hawk and Dove payoffs are then further diminished. A safer strategy of lower cost display, bluffing and waiting to win, then becomes viable – a Bluffer strategy.

This is more of a Roy situation with scarce resources, competition is better because cooperation is not possible as for example 2 birds cannot share the food source and both live. Then in an Oy-R interaction they use bluff and deception, there is no normal equilibrium but instead booms and busts.

The game then becomes one of accumulating costs, either the costs of displaying or costs of prolonged unresolved engagement. It’s effectively an auction; the winner is the contestant who will swallow the greater cost while the loser gets, for all his pains, the same cost as the winner but NO resource.[9] The resulting evolutionary game theory mathematics leads to an optimal strategy of timed bluffing.[10]

Being Roy costs need to be minimized to survive, there is no equilibrium because the lose is far worse off because the conflict cost them energy and got them nothing. Whichever reaches a tipping point then loses.

War of attrition for different values of resource. Note the time it takes for an accumulation of 50% of the contestants to quit vs. the Value(V) of resource contested for

The difference is in Aperiomics a war of attrition occurs between Y-Ro or V-Bi teams rather than chaotic fighting with bluffs, instead these wars are more open and honest. For example trench warfare in WW1 between teams of soldiers with little deception and bluffing. Sometimes single animals can use Y-Ro strategies, for example a lone Ro buffalo might confront a lone Y lion. Both cannot hide from or bluff each other, the survivor is likely to be whichever lasts the longest before becoming exhausted.

This can also occur in human games like chess or boxing where a war of attrition is open and honest, each can see what reserves of strength and endurance the other has. Sometimes though becoming weak can cause a chaotic collapse like a city at the end of a long siege. A boxer might become tired and then suddenly collapse chaotically, this happened for example with the Germans in WW1.

This is because in the war of attrition any strategy that is unwavering and predictable is unstable – unstable in the sense that such a strategy will ultimately be displaced by a mutant strategy which will simply rely on the fact that it can best the existing predictable strategy just by investing just an extra small delta of waiting resource to insure that it wins.

Here predictable means deterministic or chaotic, Oy predators and R prey each try to find patterns in each other's movements they can take advantage of. Mutant Oy-R animals can win or lose in booms and busts with no equilibrium, for example if R doves become a little faster than Oy hawks then they can feed with impunity.

Therefore only a random unpredictable strategy can maintain itself in a population of Bluffers.

In Aperiomics only a chaotic and deterministic strategy that is concealed will work. A random strategy will sometimes leave an animal exposed just by chance, for example an R gazelle that runs away randomly will sometimes run towards an Oy predator. They must then uses tactics to work out a path while obfuscating this. A random strategy works in Ro herds and with Y predators, they can afford to move randomly because they get the advantage of being unpredictable while still being open enough that they can anticipate each other's approximate movements. The teamwork then covers the defect of the R gazelle using randomness because this seals up the cracks in a team defence.

The contestants in effect choose an “acceptable cost” to be incurred related to the value of the resource being sought – randomly selected at contest start - effectively making a related random “bid” which becomes part of a mixed strategy (a strategy where a contestant has several, or even many, possible actions in his strategy).

Ro or Y animals use a random strategy which means they all accept a chance of being hurt randomly, therefore they can accept this risk as part of a cooperative team.

Asymmetries that allow new strategies

Aperiomics is a highly asymmetric system because Oy is chaotic and Ro is random in animal contests in the middle of the food chain. At the top of the food chain are Y teams such as prides of lions, then below then are loner competitive strategies such as hyenas or cheetas. Then in the middle of the food chain are O animals that are part predator and part prey, for example wild dogs might be attacked by other predators and also attack prey such as gazelles. Hyena can also be in this situation, killed and eaten by Y lions, also having their food stolen by lions. They also however act as predators for weaker animals.

Below this are the Ro herds that use cooperative behaviour against Oy loners and O hybrid predator/prey animals. Then below them are where the Ro herd strategy fails, this is R competitive prey.

In the War of Attrition there must be nothing that signals the size a bid to an opponent, otherwise this can act as a cue which can be utilised by an opponent for an effective counter-strategy for a mutant.

Usually in Y-Ro and V-Bi there is little hidden, for example chess is usually a war of attrition where no resources are hidden and the opponent is usually ground down.

There is however a mutant strategy which can better that of Bluffer in the War of Attrition Game if such an asymmetry exists. This mutant strategy is the Bourgeois strategy (Maynard Smith named the strategy Bourgeois, because with his background of communism he regarded it as “politically bourgeois” way to value ownership). Bourgeois uses an asymmetry of some sort to break the deadlock.

In Aperiomics Ro and Bi can represent the bourgeois in economics, they use the asymmetry of cooperating with each other warning of dangers.

In nature one such asymmetry is possession - which contestant has the prior possession of the resource. The strategy is to play a Hawk if in possession of the resource but to display then retreat if not in possession.

Y-Ro are positional colors that have possession of territory while Oy-R animals use momentum and speed in an asymmetrical contest. usually then the Oy-R animals rely on movement to succeed rather than possessing territory, for example Y lions might possess a territory while Oy hyena feed on it might hiding from the lions and giving up their kills if discovered. A Ro herd of Ibex might use this asymmetric strategy to stand together against O wild dogs but run from Oy hyena or Y lions.

This strategy requires greater cognitive capability than Hawk but Bourgeois is still a very prevalent strategy in many animal contests… some examples being Mantis Shrimp and Speckled Wood Butterfly contests. Thus, evolutionary game theory explains a very wide range of rather mystifying behaviours in animals and in plant contests for many species with an illuminating clarity.

Strategic alternatives in social behaviour

Alternatives for game theoretic social interaction

Games like Hawk Dove and War of Attrition represent pure competition between individuals and have no attendant social element to the game.

In Aperiomics war of attrition is a cooperative game, in war it is used by cooperating soldiers in armies.

Where social influences apply there are four possible alternatives for strategic interaction that exist for the competitors.

In a Cooperative or Mutualistic relationship both “donor” and “recipient” are almost indistinguishable as both gain a benefit in the game by co-operating, i.e. the pair are in a game-wise situation where both can gain by executing a certain strategy, or alternatively both must act in concert because of some encompassing constraints that effectively puts them “ in the same boat together”.

This is like Y-Ro and V-Bi except they need the opposing team to reinforce the need to cooperate.

In an Altruistic relationship the donor, at a cost to himself provides a benefit to the recipient. In the general case the recipient will have a kin relationship to the donor and the donation is one-way. Behaviours where benefits are donated alternatively (in both directions) at a cost, are often called altruistic, but on analysis such “altruism” can be seen to arise from optimised “selfish” strategies.

In Roy situations this can be to minimize costs, for example soldiers in an army might be willing to die for each other so more survive overall.

Spite is essentially a “reversed” form of altruism where an ally is aided by damaging the ally’s competitor(s). The general case is that the ally is kin related and the benefit is an easier competitive environment for the ally. Note: George Price, one of the early mathematical modellers of both altruism and spite, found this equivalence particularly disturbing at an emotional level.[13]

Spite can be about the need for an opposing team to inspire cooperation.

Selfishness is the base criteria of all strategic choice from a game theory perspective – strategies not aimed at self-survival and self-replication are not long for any game. Critically however, this situation is impacted by the fact that competition is taking place of multiple levels - i.e.at a genetic, an individual and a group level.

These are Iv-B and Oy-R interactions. There are also other interactions outlines in Aperiomics, for example the role of police and a justice system as I-O.

The rationale and the mathematics that lie behind adopting one of these alternative social strategies will be covered in the following sections of this article.

Who is Playing the Game?

The Belding's Ground Squirrel lives in communities of closely related females and their young and male “immigrants”. This is so, because males leave the colony on reaching maturity and find other colonies to join. When predators are in the vicinity of a colony certain squirrels emit a loud piercing alarm call, allowing other colony members to take cover. This call substantially endangers the caller as it easily locates it for the predator. However as female squirrels are so closely related Evolutionary Game Theory utilising measures of Inclusive Fitness shows that this behaviour is superior to not calling for them. Field studies confirm this is exactly how the females behave. The males however, having no such level of inclusive fitness in general do not call.[14]

This is an example of males being more R as loners while the females are more Ro cooperating with each other.

Eusociality and Kin Selection

“Ants are good citizens, they place group interests first” Clarence Day

They are a combination of R and Ro, also to some degree like the B roots and Bi upper root system of plants. Their journeys look like the roots of a plant with some cooperation at levels representing a normal curve.

Insect societies have always been a source of fascination and inscrutability. Certainly one of the very most inexplicable behaviours of these eusocial insects is the forfeiture of reproductive rights that the workers grant in favour of the queen. In a Darwinian sense, no greater sacrifice can ever exist. The explanation that has been forthcoming is in Kin Selection influences that arise from the genetic makeup of these workers which predisposes them to such kin oriented altruistic behaviours .[19] Most eusocial insect societies have haplo-diplod sexual determination, which in essence means that males develop from unfertilised eggs, females from fertilised. This leads to the situation in these Haplodiploid species, that sisters share 75% of their genes in common…. in effect more that they genetically share with their mother. But in this system ALL the other workers are sisters, further skewing the kin related payoffs toward altruistic sacrifice. Helping to raise the queens offspring, which are also sisters, therefore has major payoffs as does sacrificing onself if the nest is invaded by some predator. For those insect societies which do not exhibit haplodiplodal genetic relationships, the most significant example being termites, there still exist kin related influences arising from inbreeding, monogamy and the ineffectiveness of dispersal [20]

Ants have evolved this strategy because it works the best against Oy and Y predators, those that tried other strategies survived less often. For example more selfish ant behaviour prevented them working as a Ro team using their combined strength against isolated Oy predators.

The widely accepted kin related explanation of insect eusociality has however been challenged recently by a few highly noted evolutionary game theorists (Nowak and Wilson) [21] who have published a controversial alternative game theoretic explanation based on a sequential development and group selection effects proposed for these insect species.[22]

[edit]Prisoners Dilemma

One of the great difficulties of Darwinian Theory, and one recognised by Darwin himself was the problem of altruism. If the basis for selection is at the individual level, altruism makes no sense at all. But universal selection at the group level (for the good of the species, not the individual) fails to pass the test of the mathematics of game theory and is certainly not found to be the general case in nature.[23] Yet in many social animals altruistic behaviour can be found. The solution to this paradox can be found in the application of Evolutionary Game Theory to the “Prisoners Dilemma” game - a game which tests the payoffs of cooperating or in defecting from cooperation. It is certainly the most studied game in all of Game Theory.[24]

As with all games in Evolutionary Game Theory the analysis of Prisoners Dilemma is as a repetitive game. This repetitive nature affords competitors the possibility of retaliating for “bad behaviour” (defection) in previous rounds of the game. There is a multitude of strategies which have been tested by the mathematics of EGT and in computer simulations of contests and the conclusion is that the best competitive strategies are general cooperation with a reserved retaliatory response if necessary.[25]

Not if there is an advantage to moving first or moving faster in Iv-B and Oy-R.

The most famous and certainly one of the most successful of these strategies is Tit for Tat which carries out this approach by executing the simplest of algorithms.

The pay-off for any single round of the game is defined by the pay-off matrix for a single round game (shown in bar chart 1 below). In multi-round games the different choices - Co-operate or Defect - can be made in any any particular round, resulting in a certain round payoff. It is, however, the possible accumulated pay-offs over the multiple rounds that count in shaping the overall pay-offs for differing multi-round strategies such as Tit-for-Tat.

Routes to Altruism

The ESS

The Evolutionary Stable Strategy (ESS) is perhaps the most widely known albeit most widely misunderstood concept in Evolutionary Game Theory. The ESS is basically akin to Nash Equilibrium in classical Game Theory, but with mathematically extended criteria.

The Payoff Matrix for the Hawk Dove Game with and additional strategy - the Assessor Strategy. The Assessor competitor "studies its opponent" and behaves as a Hawk when matched with an opponent it judges "weaker", and behaves like a Dove when it assesses its opponent as stronger. (The usual judgement criteria in nature is opponent size) Assessor is a ESS. As can be seen in the payoffs of the matrix it can invade as a mutant into both Hawk and Dove populations, and can ALSO withstand invasion by either Hawk or Dove mutants as it gets a better payoff when matched against another Assessor

This acts as the O middle of the food chain in Roy systems and I in Biv systems.

Nash Equilibrium is a game equilibrium where it is not rational for any player to deviate from the present strategy they are executing. As discussed, in Evolutionary game Theory contestants are NOT behaving with rational choice, nor do they have the ability to totally alter their strategy, aside from executing a very limited “mixed strategy”. An ESS is instead a state of game dynamics where, in a very large (or infinite) population of competitors, another mutant strategy cannot successfully enter the population to disturb the existing dynamic (which in itself is population mix dependent). This leads to a situation where to be a successful strategy having an ESS, the strategy must be both effective against competitors when it is rare - to enter the previous competing population, and also successful when later in high proportion in the population - to “defend itself”.[31] This in turn necessarily means that the strategy needs to be successful when it contends with others exactly like itself.

ESS is NOT:

An OPTIMAL strategy – an optimal strategy would maximize Fitness, and many ESS states are far below the maximum fitness achievable in a fitness landscape. (see Hawk Dove graph above as an example of this)

A singular solution – often several ESS conditions can exist in a competitive situation. A particular contest might stabilize into any one of these possibilities, but later a major perturbation in conditions can move the solution into one of the alternative ESS states.

Always present - It is also possible for there to be no ESS. An example evolutionary game with no ESS is the Rock-Scissors-Paper game found in a number of species (an example the side-blotched lizard (Uta stansburiana))

An unbeatable strategy - The ESS strategy is not necessarily an unbeatable strategy, it is only an uninvadable one.

An Assessor Strategy Player - - Female funnel web spiders (Agelenopsis aperta) contest with one another for the possession of their desert spider webs. A higher quality established web site offers a considerable reproductive advantage and is well worth holding, but serious fights for them are potentially very dangerous for the contestants. The spiders have been found to use the Assessor strategy, with size the critical determinant of winner. When an intruder spider enters a web to contest for possession the spiders become involved in a “web bouncing” behaviour which in essence establishes size by "weighing" the two contestants. The smaller spider, even if the original possessor of the web, leaves the site of the contest. In a detailed study Riechert proved that an Assessor "weighing" strategy was in play by adding an additional ballast weight to physically smaller spiders.[32]

They in effect evolve a primitive I-O justice system, instead of fighting there is a negotiation. O animals assess whether they are more likely to be prey or predator when they encounter other animals.

The ESS state can be solved for mathematically by exploring either the dynamics of population change to determine any ESS.... or alternatively by solving equations for the stable stationary point conditions which fundamentally define an ESS.[33] For example, in the Hawk Dove Game we can look for whether there is a static population mix condition where the fitness of Doves will be exactly the same as fitness of Hawks (therefore both having equivalent growth rates - a "static point").

Similarly using inequalities it can be shown that an additional Hawk or Dove “mutant” entering this ESS state generates a situation leading eventually to LESS fitness for their kind – both a true Nash and an ESS equilibrium. This fairly simple example shows that when the risks of contest injury or death (the Cost C) is significantly greater than the potential reward offered (the benefit value V) then the stable population which is reached will be MIXED between the aggressors and the doves, and that the proportion of doves will exceed that of the aggressors. This then mathematically explains behaviours that are actually observed in nature.

This can lead to where I and O are biased more to the left as prey or right as predator making the system more unstable.

[edit]Rock Scissors Paper Game

Rock Scissors Paper - -

An evolutionary game that actually turns out to be a children’s game is rock-paper-scissors. The game is simple – rock bests scissors (blunts it), scissors bests paper (cuts it), and paper bests rock (wraps it up). Anyone who has ever played this simple game knows that it is not sensible to have any favoured play – your opponent will soon notice this and switch to the winning counter-play. The best strategy (a Nash equilibrium) is to play a mixed random game with any of the three plays taken a third of the time. This, in EGT terms, is a mixed strategy. But many lifeforms are incapable of mixed behavior — they only exhibit one strategy (known as a “pure” strategy”). If the game is played only with the pure Rock, Scissor and Paper strategies the evolutionary game is dynamically unstable: Rock mutants can enter an all scissor population, but then – Paper mutants can take over an all Rock population, but then – Scissor mutants can take over an all Paper population – and on and on…. This is easily seen on the game payoff matrix, where if the paths of mutant invasion are noted, it can be seen that the mutant "invasion paths" form into a loop. This in triggers a cyclic invasion behaviour.

This can create virtuous and vicious cycles in Iv-B and Oy-R.

Rock Paper Scissors Dynamics A computer simulation of the Rock Scissors Paper evolutionary game. The associated RPS Game Payoff Matrix is shown. Starting with an arbitrary population the percentage of the three morphs builds up into a continuously cycling pattern – with rock leading paper, paper leading scissors, and scissors leading rock etc. etc.

[edit]The Side-Blotched Lizard

The Side-Blotched Lizard (Uta stansburiana) is polymorphic with three morphs[34] that each pursues a different mating strategy

Uta Stansburiana — the Side Blotched Lizard

1) The Orange throat is very aggressive and operates over a large territory - attempting to mate with numerous females within this larger area

2) The unaggressive Yellow Throat (called “sneakers”) mimic the markings/behavior of female lizards and sneakily slip into the Orange Throat's territory to mate with the females there (thereby overtaking the population), and

3) The Blue Throat who mates with and carefully guards ONE female - making it impossible for the sneakers to succeed and therefore overtakes their place in a population…

However the BlueThroats cannot overcome the more aggressive Orange Throats… And on, and on… The overall situation corresponds to the exact form of the Rock, Scissors, Paper game and the dynamics is much the same. The populations for these lizards actually cycle on a six year basis. Once again EGT explains a very curious and otherwise inexplicable behaviour in the field of biology. When he read that lizards of the species Uta stansburia were essentially engaged in a game with rock-paper-scissors structure John Maynard Smith exclaimed `They have read my book!´[35]

RPS and Ecology

Rock Scissors Paper is an evolutionary games that also helps to develop an understanding of Ecology.[36] In two strategy “pairwise” contests the outcome in most contests is that one strategy dominates the population and the dynamic reaches a static ESS point. This means in ecological terms, where it is species that compete with one another, that one particular species in these pairwise contests tends to dominate a particular ecological niche. Most ecologies however are built up of multiple species (each successful in a separate niche), in intertwined or “intransitive” relationships, where each species benefits indirectly from a competition taking place between two other species, and where many species populations co-exist with a degree of cycling. These multi species ecologies have been modelled by game-theoretic analysis based on the Rock Scissors Paper game and results have been shown to produce dynamics that closely match with ecological systems in nature.

This also occurs in economies where Biv companies might form virtuous circles where each profits from this nontransivity. However when the economy turns to Roy as in the GFC they can become vicious circles where in minimizing losses they drag each other down. In the example above when the environment became poor enough as Roy they would each tend to attack the next one in the circle causing it to collapse chaotically.

Signalling and the Handicap Principle

The Handicap Principle in action

Aside from the difficulty of explaining how altruism exists in many evolved organisms Darwin was also bothered by a second conundrum –why do a significant number of species have phenotypical attributes that are patently disadvantageous to them with respect to their survival - and should by the process of natural section be selected against – e.g. the massive inconvenient feather structure found in peacocks. tails? Regarding this issue Darwin wrote to a colleague “ The sight of a feather in a peacocks tail, whenever I gaze at it, makes me sick”.[37] It is the mathematics of Evolutionary Game Theory, which has not only explained the existence of altruism but also explains the totally counterintuitive existence of the peacock’s tail and other such biological encumbrances.

On analysis, problems of biological life are not at all unlike the problems that define economics – eating (akin to resource acquisition and management), survival (competitive strategy) and reproduction (investment, risk and return). Game theory was originally conceived as a mathematical analysis of economic processes and indeed this is why it has proven so useful in explaining so many biological behaviours. One important further refinement of the EGT model that has particular economic overtones rests on the analysis of COSTS. A simple model of cost assumes that all competitors suffer the same penalty imposed by the Game costs, but this is not the case. More successful players will be endowed with or will have accumulated a higher “wealth reserve” or “affordability” than less successful players. This wealth effect in Evolutionary Game Theory is represented mathematically by “resource holding potential (RHP)” and shows that the effective cost to a competitor with higher RHP are not as great as for a competitor with a lower RHP. As a higher RHP individual is more desirable mate in producing potentially successful offspring, it is only logical that with sexual selection RHP should have evolved to be signalled in some way by the competing rivals, and for this to work this signally must be done honestly. Amotz Zahavi has developed this thinking in what is known as the Handicap Principle,[38] where superior competitors signal their superiority by a costly display. As higher RHP individuals can properly afford such a costly display this signalling is inherently honest, and can be taken as such by the signal receiver. Nowhere in nature is this better illustrated than in the magnificent and costly plumage of the male Peacock. The mathematical proof of the handicap principle was developed by Alan Grafen using Evolutionary Game Theoretic modelling.[39]

It can also be a chaotic deception, peacocks are individuals rather than working as a team and so must compete against each other for mates. The tail feathers become increasingly deceptive as they do not accurately correspond to the fitness of peacocks to survive against predators. The larger the tail the more chance they might get eaten. So the female is in effect an R prey and the peacock an Oy predator in the sense that it uses deception to survive by breeding instead of eating the prey. Because this is an advantage to both the male and female it can also be a Biv game of deception, the female as B and the male as Iv being like branches of a tree having flowers and beautiful leaves. The males then use V plumage like trees do, to overshadow rivals who cannot compete with it.

Co-Evolution

Two types of dynamics have been discussed so far in this article:

Evolutionary games which lead to a stable situation or point of stasis for contending strategies which result in an Evolutionary Stable Strategy

Evolutionary games which exhibit a cyclic behaviour (as with RPS game) where the proportions of contending strategies continuously cycle over time within the overall population

Competive Coevolution

- The Rough-skinned newt (Tarricha granulosa) has enough poison in its body to kill a human being 30 times over. This excess of lethal capability is due to the fact that it is involved in an arms race with a specific predator, the common garter snake, which in response to the newts poisonous defenses has over time has evolved to be highly tolerant of the poison. The two species are thereby locked in a "Red Queen" arms race.[40]

This is chaotic Oy-R where each R innovation of creating poison leads to a counter innovation in Oy of tolerating it. This is like R victims of crimes innovating new ways of protecting themselves and Oy thieves counter innovating to overcome these.

Mutualistic Coevolution

- Darwins orchid(Angraecum sesquipedale)and the moth Morgan's Sphinx (Xanthopan morgani) like many insect and flower "partners" have a mutual relationship where the moth gains pollen and the flower pollination. To insure efficiency in this particular exchange the pair have evolved a mechanism which excludes the waste of pollen transfer to/from another flower species and assures feeding pollin only to the "proper" pollinator. The mechanism is an extraordinarily long proboscis on the moth and a equally long neckery on the orchid.

V parts of plant cooperate with each other and R prey such as the moths are chaotic, however sometimes random cooperation can be stronger than chaotic competition. The R moths get more food by not chaotically damaging the flowers, this is like R gazelles evolving to not damage the grass they eat and so having a larger food supply.